Single-electron charge transfer into putative Majorana and trivial modes in individual vortices

Majorana bound states are putative collective excitations in solids that exhibit the self-conjugate property of Majorana fermions—they are their own antiparticles. In iron-based superconductors, zero-energy states in vortices have been reported as potential Majorana bound states, but the evidence remains controversial. Here, we use scanning tunneling noise spectroscopy to study the tunneling process into vortex bound states in the conventional superconductor NbSe2, and in the putative Majorana platform FeTe0.55Se0.45. We find that tunneling into vortex bound states in both cases exhibits charge transfer of a single electron charge. Our data for the zero-energy bound states in FeTe0.55Se0.45 exclude the possibility of Yu–Shiba–Rusinov states and are consistent with both Majorana bound states and trivial vortex bound states. Our results open an avenue for investigating the exotic states in vortex cores and for future Majorana devices, although further theoretical investigations involving charge dynamics and superconducting tips are necessary.

Then we perform tunneling spectroscopy with this tip on a clean Au(111) surface at various magnetic fields, as shown in Supplementary Fig. 1b. The differential conductance measured by tunneling spectroscopy is expressed by where E is energy, Ns (Nt) is the density of states (DOS) in the sample (tip), and f (E, T) = [1+exp(E/kBT)] -1 is the Fermi function (kB being the Boltzmann constant). We model the tip DOS by the Dynes function [1] t ( , , Γ) = Re [ where Δt is the superconducting energy gap of the tip and Γ is the phenomenological broadening parameter (without thermal broadening). The values for Δt and Γ are extracted by fitting each spectrum to Supplementary Equation 1, assuming a constant Ns for Au(111) in the energy range from -10 meV to +10 meV. The fit results are summarized in Supplementary Fig.  1c, showing a critical field of 0.7 T, about 7 times larger than that of the bulk Pb. [2] Supplementary Note 2: Deconvolution of the conductance spectra taken with a superconducting tip The consequence of using a superconducting tip with an energy gap Δt, as illustrated by Supplementary Fig. 2a, is the resonance tunneling when either of the gap edges of the tip DOS aligns with the zero-energy state (ZES). Thus, the bound state appears as peaks at ±Δt in the measured differential conductance spectra. Deconvolution of a differential conductance spectrum is necessary to recover the appearance of the bound state at zero energy in the sample DOS, the same as what one would expect for a spectrum taken with a normal-metal tip. [3,4] We follow the deconvolution algorithm described in Refs. [5,6] to extract the sample DOS Ns in Supplementary Equation 1. We use the fit results at B = 0.1 T in Supplementary Fig. 1c for the tip DOS Nt in the deconvolution.
To determine the energy of the vortex bound states in FeTe0.55Se0.45, in Supplementary Fig. 2b, c we stack the raw and deconvoluted spectra for the line cut in Fig. 3c, f, respectively. We fit each peak (at ±Δt for Supplementary Fig. 2b and at 0 meV for Supplementary Fig. 2c) with a Lorentzian function, and plot the peak energy as a function of the position along the line in Supplementary Fig. 2d. We note an additional broadening of the coherence peaks and the zerobias peak in the deconvoluted DOS (Fig. 3e) compared to the peaks in the raw spectrum in Fig.  3b, due to the need in the deconvolution algorithm to remove oscillatory errors (we chose the optimized value for the control parameter  = 5.0 defined in Ref. [5]). Nevertheless, the peak center locates at 0 ± 50 μeV, confirming them as zero-energy states. On the other hand, from the amplitudes of the Lorentzian fit ( Supplementary Fig. 2e), we find the zero-energy state has roughly an exponential decay in DOS, with a decay length of ~ 4.0 nm on both sides of the core center.

Supplementary Note 3: Differential conductance and noise spectroscopy on different vortices
We present the full dataset of both differential conductance and noise spectroscopy performed on all vortices in Supplementary Figs Fig. 4d is due to a higher junction resistance of 5 MOhm (and 10 MOhm) used for the noise spectroscopy measurements, leading to reduced absolute values of tunnel current and its noise.

Supplementary Note 4: Scanning noise spectroscopy at zero field around a YSR impurity
The results of our measurements on FeTe0.55Se0.45 at zero magnetic field are shown in Supplementary Fig. 6. Here we observe the Yu-Shiba-Rusinov (YSR) bound states as a ring in the differential conductance map ( Supplementary Fig. 6b). The YSR states lead to a negative differential conductance in the spectrum ( Supplementary Fig. 6d) because of the convolution of a sharp in-gap resonance peak and superconducting tip DOS as we observed in a previous study. [6] The noise spectrum measured away from the impurity site ( Supplementary Fig. 6e) shows clear transitions from q * = 1e line to q * = 2e line with onsets at ±(Δt +Δs), indicating a dominating Andreev reflection inside the gap. This noise behavior is similar to what we observed before on Pb(111) surface with a superconducting tip. [7] We extract the effective charge q * in Supplementary Fig. 6f by numerically solving Eq. 2 in the main text. We observe a narrower step of q * from 1e outside the gap to 1.97e at E = ±Δt, compared to the broader transition from 1e to a plateau of 1.3e~1.6e in Supplementary Fig. 5. The value of q * so close to 2e in Supplementary Fig. 6f indicates the tunneling current originates purely from Andreev reflection in the SIS junction, whereas q * short of 2e in Supplementary Fig. 5 off vortex implies that the tunneling process is not purely Andreev reflection, and that a contribution from 1echarge tunneling coexists.
A previous study [8] demonstrated, by both theory and experiment, that Andreev process dominates single-particle tunneling into YSR states in the strong tunneling limit Γ1 << Γt, where Γt is the tunneling rate and Γ1 is the threshold rate for quasiparticles to be excited into the continuum. For our experiment on FeTe0.55Se0.45, we estimate Γ1 = 1.1 μeV from Eq. (S49) of Ref. [8], using Δs = 1.5 meV, T = 2.3 K, and the YSR energy of 0.3 meV. This yields a threshold current of ~ 90 pA from Eq. (S65) of Ref. [8]. We expect that, in this limit, the effective charge when tunneling into YSR states is identical to that into the bare superconductor at a bias energy inside the gap.
We have verified in the strong tunneling limit (with a current of 800 pA) that the YSR state does not cause a spatial difference in shot noise, as shown by the noise map in Supplementary  Fig. 6c, compared to the prominent ring feature in the differential conductance map taken in the same field of view and at the same bias voltage ( Supplementary Fig. 6b). Therefore, we exclude YSR states as the origin of the zero-energy bound states in the vortex cores of FeTe0.55Se0.45.
Note that there is no contradiction between Ref. [9] and our data for YSR states. For tunneling into the bare superconductor NbSe2, Ref. [9] measures 1e noise outside the gap but lacks of noise data inside the gap because of the low current (~20 pA), for which they could not detect the corresponding shot noise accurately. In principle, considering a normal tip used in this case, inside the gap q * increases gradually to 2e [10]. For tunneling into YSR states, Ref. [9] elucidates that Andreev reflection requires particle-hole symmetric resonances. The smaller of the asymmetric resonances leads to a deficiency of providing particle (or hole) for the Andreev (2e) process, so that tunneling into the excess hole (or particle) component in the larger resonance has to be mediated by an inelastic quasiparticle relaxation process. In our experiment this inelastic quasiparticle relaxation (1e) process is strongly suppressed because the YSR states in FeTe0.55Se0.45 are almost symmetric in amplitude and our noise measurements were carried out in the strong tunneling regime. In summary, both our data and Ref. [9] reach the conclusion that tunneling into YSR states, when inelastic quasiparticle relaxation can be neglected, gives 2e noise.

Supplementary Note 5: The effective charge when single-particle and Andreev processes both contribute
In this section, we calculate, based on an empirical model, the effective charge when both Andreev reflection and quasiparticle of 1e tunneling contribute to the total current. Away from the vortex, the tunnel junction is similar to an SIS junction, as shown by comparing the spectra in Fig. 2b and Fig. 3b and the zero-field spectrum in Supplementary Fig. 6d. Especially at the bias energy E = ±Δt, the deconvolution yields a vanishing density of states of the sample (Figs. 2e and 3e). Therefore, the tunneling process for this tunnel junction is expected to be dominated by Andreev reflection that transfers a charge of 2e per event. We then introduce a fraction of the 1e-charge tunneling process, [11] which contributes to current and noise but has no correlation with those of the Andreev process. For a given tunneling transparency  << 1, the current contributions for the single-particle processes (I1e) and the Andreev processes (I2e) are proportional to  and  2 , respectively, [12] Ine  n n /4 n-1 , n = 1, 2. (S3) The prefactors are related to the integrated density of states, and here we assume an empirical prefactor y for quasiparticle contribution I1e and 1-y for I2e to have a conserved total integrated density of states. The total current is I = I1e + I2e. As I1e and I2e are assumed to be independent, the total current noise is the sum of both contributions, where the double-charge (2e) transfer is taken into account in the Andreev contribution (the second term). Then we extract numerically the (total) effective charge q * by Eq. 2.
Supplementary Figure 7 plots q * as a function of the fraction of quasiparticle contribution for different junction resistance we used in noise measurements. When I1e/I = 0, i.e., no singleparticle process contributes, q * = 2e as expected from purely Andreev reflection. Conversely when I1e/I = 100%, only single-particle process contributes, yielding q * = 1e. For values of I1e/I in between 0 and 100%, we find a quick reduction of q * even when a very small fraction of quasiparticle contribution exists (note the logarithmic scale of the horizontal axis). For example, for RJ = 2.5 MOhm, 0.02% of quasiparticle contribution reduces q * to 1.92e, while 3.3% of quasiparticle contribution reduces already reduces q * to 1.07e.

Supplementary Note 6: Transparency of the tunnel junction during noise spectroscopy
In differential conductance (Figs. 2 and 3) and noise (Fig. 4) spectroscopy, we use different setup conditions in terms of feedback control of the tip. Specifically, in differential conductance measurements, as the protocols are conventionally applied, feedback is disabled during voltage sweeps (spectroscopy). However, in noise spectroscopy, in order to have optimal junction stability, we enable a slow feedback to maintain a constant junction resistance (i.e., changing the bias voltage and current setpoint for each point in a sweep), except at the zero-bias point where feedback has to be disabled. As already presented in Fig. 2b, the differential conductance of an SIS junction varies by an order of magnitude during a sweep, the transparency of the junction, if feedback is enabled, could also vary considerably. The transparency , which is assumed to be in the  << 1 limit for current noise expressions in the main text, has a significant influence on the resulted noise when it becomes comparable to unity. [9] In Supplementary Fig. 8a we compare the differential conductance taken with feedback disabled and enabled, at different junction resistance (thus the setup ). While outside the gap the conductance measured both ways is almost identical, a drastic difference in conductance develops in the gap because of a vanishing quasiparticle density of states. The ratio of the feedback-on conductance over the feedback-off conductance gon/goff indicates the enhancement of transparency from  = (RJG0) -1 from an Ohmic current-voltage relation, where G0 = 2e 2 /h = 77.5 μS is the conductance quantum (h being the Planck constant). In fact, the conductance ratio gon/goff is barely dependent on RJ, as shown in Supplementary Fig. 8b Supplementary Fig. 8b), in good agreement with the experimental results. From Supplementary Fig. 8 we have confirmed that in our measurement conditions, enabling feedback has a marginal effect on the noise, as the transparency  stays below 0.16, which is still in the  << 1 limit.

Supplementary Note 7: Different dispersions of CdGM states in NbSe2 and FeTe0.55Se0.45
For a conventional s-wave BCS superconductor, CdGM states have been extensively studied theoretically (e.g., Ref. S11): the CdGM states have an increasing angular momentum when moving a distance r away from the vortex core. As a consequence, the majority of the CdGM states that contributes to the differential conductance have an energy Ep approximately proportional to kF⋅r, where kF is the Fermi wavevector. In addition, at Ep the differential conductance maximum decays exponentially in r on a length scale of coherence length ξ. Therefore, the dispersion profile of CdGM states depends crucially on two material parameters kF and ξ (see Supplementary Table 1 for their values of NbSe2 and FeTe1-xSex). For NbSe2, both parameters are larger, and the dispersion is measurable by STM. However, for FeTe1-xSex, kF is one order of magnitude smaller, so Ep changes much more slowly with r; meanwhile ξ is also smaller, resulting in a vanishing amplitude in differential conductance before Ep changes significantly. Therefore, observation of dispersing states indicates a CdGM origin, but nondispersing states cannot exclude a CdGM origin.

xSex.
Material NbSe2 FeTe1-xSex Fermi wavevector kF 0.5~1 Å -1 [12] 0.07~0.12 Å -1 [13] Coherence length ξ 12 nm [14] 3 The effective charge numerically extracted from Fig. 4a and Fig. 4c, respectively, by Eq. 2 in the main text. Error bars correspond to numerical solutions using values of upper and lower bounds indicated the error bars in Fig. 4a and Fig. 4c, respectively. A further increase of q * above within 0.4 meV is caused by an increasing Andreev (and possible multiple Andreev) contribution because of a vanishing tip DOS. Unfortunately, the uncertainty also increases quickly for the numerical solution within this range due to the divergence of the coth function when Vbias → 0 in Eq. 2 in the main text.
Supplementary Figure 6. Noise measurements near a YSR impurity at zero magnetic field. a,b STM topography (a) and differential conductance (b) for bias Vbias = -2 mV for the same field of view (20 nm × 20 nm). A ring feature with enhanced (reduced) conductance outside (inside) indicates that resonant tunneling occurs when approaching the impurity. c Grid spectroscopic map of noise measured near the impurity (the green square in b) at the same bias Vbias = -2 mV. The ring feature is absent in noise. d Differential conductance spectra measured on and off the impurity, denoted by the red and blue crosses in b, respectively. e,f Noise spectrum (e) and its corresponding effective charge ( Fig. 4 are taken. Error bars are determined by the fluctuation of the current noise in time, yielding a standard deviation of 9.25 fA 2 /Hz. The lines show theoretical expectations of shot noise, for q * =1e (black), q * =2e (gray), and q * =2e with correction for nonlinear junction transparency (black dotted line).
Supplementary Figure 10. Differential conductance and noise spectra measured on NbSe2 at zero field. a Differential conductance spectrum and b corresponding noise spectrum measured at B = 0 T and a random location on a NbSe2 sample. Noise data increase from q * =1e curve starting at ±(Δt + Δs), towards to q * =2e curve inside the gap. Setup conditions: a, Vset = -5 mV, Iset = 200 pA; b, RJ = 2.5 MOhm.